## Thursday, June 21, 2012

### The basics of scientific notation

Scientific notation can be used to represent any positive number. Usually, we only use it for really small numbers (close to zero) or really big numbers (more than a billion), but the method will work for anything.

Scientific notation is the form a x 10^b, where a is a number between 1 and 10 (1 is included but 10 isn't) and b is an integer, which means a whole number or the negative of a whole number.

The letter a is called the significand and b is the exponent or sometimes the order of magnitude.

Example with a "regular sized" number.

The number of feet in a mile is 5,280.  I want to write it as a number between 1 and 10 times a power of 10. 5,280 is a four digit number, so the nearest power of 10 is 1,000 or 10^3. If I divide 5280/1000, I get 5.28.  The mechanical way to do this is to move the decimal place on 5280. over three places to the left.

5280. =
528.0 x 10^1 =
52.80 x 10^2 =
5.280 x 10^3 or 5.28 x 10^3 (Your calculator will not write trailing zeros after a decimal.)

The last line is scientific notation because 1 < 5.28 < 10. 5280, 528 and 52.8 are too big to be significands.

One foot is 1/5280 of a mile. Type this into your calculator and you get 0.000189394..., a decimal which goes on beyond our calculators limits.  Let's round this to 0.000189.  This is called rounding to three significant digits, which means how many digits we have after we get past the leading zeros.

Because this is less than one, we are going to have to multiply by 10 raised to a negative exponent to get the scientific notation.

0.000189 =
0.00189 x 10^-1 =
0.0189 x 10^-2 =
0.189 x 10^-3 =
1.89 x 10^-4

We would say this number in scientific notation as "one point eight nine times ten to the negative fourth".

Your calculator only has ten places for digits, so a number like 25,000,000,000 (twenty five billion) is too big to be written in regular notation. If you type it in, the TI-30XIIs will display 2.5 x 10^10, or "two point five times ten to the tenth power".

Multiplying numbers in scientific notation

If I have two numbers in scientific notation, here is how I multiply them together.

1) Multiply the significands.
3) If the product from step 1 is more than 10, divide it by ten (move the decimal place one to the left) and add 1 to the exponent.

Example.

(6.1 x 10^7) x (4.3 x 10^9)  This is 61,000,000 x 4,300,000,000.

1) 6.1 x 4.3 = 26.23
2) 7 + 9 = 16
3) This gives us 26.23 x 10^16. 26.23 is too big to be a significand, so we divide it by 10 and multiply 10^16x10, which gives us 10^17.  The answer in scientific notation is 2.623 x 10^17.